Bottom Line:
In the analysis, the OSM is used first to separate the Laplacian operator and the nonlinear source term, and then the second-order time-stepping schemes are employed for approximating two splitting operators to convert the original governing equation into a linear nonhomogeneous Helmholtz-type governing equation (NHGE) at each time step.Subsequently, the RBF interpolation and the MFS involving the fundamental solution of the Laplace equation are respectively employed to obtain approximated particular and homogeneous solutions of the nonhomogeneous Helmholtz-type governing equation.Furthermore, the sensitivity of the coefficients in the cases of a linear and an exponential relationship of TDBPR is investigated to reveal their bioheat effect on the skin tissue.

Affiliation: Research School of Engineering, Australian National University, Acton, ACT 2601, Australia. zewei.zhang@anu.edu.au.

ABSTRACTA meshless numerical scheme combining the operator splitting method (OSM), the radial basis function (RBF) interpolation, and the method of fundamental solutions (MFS) is developed for solving transient nonlinear bioheat problems in two-dimensional (2D) skin tissues. In the numerical scheme, the nonlinearity caused by linear and exponential relationships of temperature-dependent blood perfusion rate (TDBPR) is taken into consideration. In the analysis, the OSM is used first to separate the Laplacian operator and the nonlinear source term, and then the second-order time-stepping schemes are employed for approximating two splitting operators to convert the original governing equation into a linear nonhomogeneous Helmholtz-type governing equation (NHGE) at each time step. Subsequently, the RBF interpolation and the MFS involving the fundamental solution of the Laplace equation are respectively employed to obtain approximated particular and homogeneous solutions of the nonhomogeneous Helmholtz-type governing equation. Finally, the full fields consisting of the particular and homogeneous solutions are enforced to fit the NHGE at interpolation points and the boundary conditions at boundary collocations for determining unknowns at each time step. The proposed method is verified by comparison of other methods. Furthermore, the sensitivity of the coefficients in the cases of a linear and an exponential relationship of TDBPR is investigated to reveal their bioheat effect on the skin tissue.

ijms-16-02001-f010: Sensitivity of temperature to constant a1 in the exponential case of blood perfusion rate.

Mentions:
In this section, the sensitivity analysis of temperature to the constant coefficients is investigated by considering the exponential case of temperature-dependent blood perfusion rate . When the constant is assumed to be 0.01, the constant is respectively tested at 0.005, 0.0005, and 0.00005. Compared with the linear case shown in Figure 8, the difference or gap between each skin temperature curve is relative larger, as shown in Figure 10. Similarly, the three temperature curves with different values of constant intersect at almost the same point (the distance from the left hand side boundary being roughly 13.125 mm). This finding means that, at the location (13.125 mm, 0), the skin temperature has almost the same value of 37.75 °C for different values of the constant . Figure 10 illustrates the stronger regulatory and protective effect of the exponential-form blood perfusion rate than that in the linear case (see Figure 8).

ijms-16-02001-f010: Sensitivity of temperature to constant a1 in the exponential case of blood perfusion rate.

Mentions:
In this section, the sensitivity analysis of temperature to the constant coefficients is investigated by considering the exponential case of temperature-dependent blood perfusion rate . When the constant is assumed to be 0.01, the constant is respectively tested at 0.005, 0.0005, and 0.00005. Compared with the linear case shown in Figure 8, the difference or gap between each skin temperature curve is relative larger, as shown in Figure 10. Similarly, the three temperature curves with different values of constant intersect at almost the same point (the distance from the left hand side boundary being roughly 13.125 mm). This finding means that, at the location (13.125 mm, 0), the skin temperature has almost the same value of 37.75 °C for different values of the constant . Figure 10 illustrates the stronger regulatory and protective effect of the exponential-form blood perfusion rate than that in the linear case (see Figure 8).

Bottom Line:
In the analysis, the OSM is used first to separate the Laplacian operator and the nonlinear source term, and then the second-order time-stepping schemes are employed for approximating two splitting operators to convert the original governing equation into a linear nonhomogeneous Helmholtz-type governing equation (NHGE) at each time step.Subsequently, the RBF interpolation and the MFS involving the fundamental solution of the Laplace equation are respectively employed to obtain approximated particular and homogeneous solutions of the nonhomogeneous Helmholtz-type governing equation.Furthermore, the sensitivity of the coefficients in the cases of a linear and an exponential relationship of TDBPR is investigated to reveal their bioheat effect on the skin tissue.

Affiliation:
Research School of Engineering, Australian National University, Acton, ACT 2601, Australia. zewei.zhang@anu.edu.au.

ABSTRACTA meshless numerical scheme combining the operator splitting method (OSM), the radial basis function (RBF) interpolation, and the method of fundamental solutions (MFS) is developed for solving transient nonlinear bioheat problems in two-dimensional (2D) skin tissues. In the numerical scheme, the nonlinearity caused by linear and exponential relationships of temperature-dependent blood perfusion rate (TDBPR) is taken into consideration. In the analysis, the OSM is used first to separate the Laplacian operator and the nonlinear source term, and then the second-order time-stepping schemes are employed for approximating two splitting operators to convert the original governing equation into a linear nonhomogeneous Helmholtz-type governing equation (NHGE) at each time step. Subsequently, the RBF interpolation and the MFS involving the fundamental solution of the Laplace equation are respectively employed to obtain approximated particular and homogeneous solutions of the nonhomogeneous Helmholtz-type governing equation. Finally, the full fields consisting of the particular and homogeneous solutions are enforced to fit the NHGE at interpolation points and the boundary conditions at boundary collocations for determining unknowns at each time step. The proposed method is verified by comparison of other methods. Furthermore, the sensitivity of the coefficients in the cases of a linear and an exponential relationship of TDBPR is investigated to reveal their bioheat effect on the skin tissue.